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Chapter 2: Problem 110

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and (3,4) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)

Short Answer

Expert verified
The figure is a parallelogram and also a rectangle.

Step by step solution

01

Calculate the slope of each side

First, calculate the slope of the line segments formed by the vertices. Use the slope formula \[ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \].
02

Calculate the slope for vertices \((-11,-5)\) and \((-2,-19)\)

Plugging the values into the slope formula:\[ m_{1} = \frac{-19 + 5}{-2 + 11} = \frac{-14}{9} = -\frac{14}{9} \].
03

Calculate the slope for vertices \( (-2,-19) \) and \((12,-10)\)

Use the slope formula:\[ m_{2} = \frac{-10 + 19}{12 + 2} = \frac{9}{14} = \frac{9}{14} \].
04

Calculate the slope for vertices \( (12,-10) \) and \((3,4)\)

Again using the slope formula:\[ m_{3} = \frac{4 + 10}{3 - 12} = \frac{14}{-9} = -\frac{14}{9} \].
05

Calculate the slope for vertices \( (3,4) \) and \((-11,-5)\)

Finally, using the slope formula:\[ m_{4} = \frac{-5 - 4}{-11 - 3} = \frac{-9}{-14} = \frac{9}{14} \].
06

Determine if the figure is a parallelogram

Since the opposite sides have equal slopes (\( m_{1} = -\frac{14}{9} \) and \( m_{3} = -\frac{14}{9} \), and \( m_{2} = \frac{9}{14} \) and \( m_{4} = \frac{9}{14} \)), the figure is a parallelogram.
07

Check for right angles to see if it's a rectangle

For the figure to be a rectangle, adjacent sides must be perpendicular. Multiply the slopes of adjacent sides to see if the product is -1.\( m_{1} \cdot m_{2} = -\frac{14}{9} \cdot \frac{9}{14} = -1 \). This condition is satisfied, thus confirming that the figure has right angles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallelogram Properties
A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. This means that if you calculate the slopes of opposite sides and they are equal, the figure is a parallelogram. Here鈥檚 what you need to know about parallelogram properties in detail:
  • Opposite sides are parallel and have equal slopes.
  • Opposite sides are equal in length.
  • Opposite angles are equal.
  • The diagonals bisect each other.
  • Adjacent angles add up to 180 degrees.
In the given exercise, the slopes of opposite sides were calculated and found to be equal. For example, the slope of vertices \((-11,-5)\) and \((-2,-19)\) was equal to the slope of vertices \((12,-10)\) and \((3,4)\). This confirms that the figure is indeed a parallelogram.
Rectangle Properties
A rectangle is a special type of parallelogram. It has all the properties of a parallelogram but with the additional feature that all angles are right angles (90 degrees). Here are the key properties:
  • All angles are 90 degrees.
  • Opposite sides are parallel and equal in length.
  • The diagonals are equal in length.
To determine if a parallelogram is a rectangle, you need to check if the adjacent sides are perpendicular to each other. This is done by multiplying the slopes of adjacent sides. If the product is -1, the sides are perpendicular. In the exercise, we found that \(-\frac{14}{9}\) multiplied by \(\frac{9}{14}\) is \(-1\), confirming that the figure has right angles and thus is a rectangle.
Slope Formula
The slope of a line measures its steepness and is calculated using the slope formula. The formula for finding the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula is essential for determining whether lines are parallel or perpendicular:
  • If two lines are parallel, their slopes are equal.
  • If two lines are perpendicular, the product of their slopes is -1.
In the exercise, we applied the slope formula to find the slopes of various segments. This helped us establish that opposite sides were parallel (equal slopes) and adjacent sides were perpendicular (product of slopes = -1), concluding that the figure is both a parallelogram and a rectangle.

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Most popular questions from this chapter

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ -2 x+5 y=9 $$

Graph each line passing through the given point and having the given slope. (0,0)\(; m=\frac{1}{5}\)

Rental on a car is \(\$ 150,\) plus \(\$ 0.50\) per mile. Let \(x\) represent the number of miles driven and \(f(x)\) represent the total cost to rent the car. (a) Write a linear function that models this situation. (b) Find \(f(250)\). Interpret the answer in the context of this problem. (c) Find the value of \(x\) if \(f(x)=400\). Express this situation using function notation, and interpret it in the context of this problem.

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ y+2 x^{2}=3 $$

A taxicab driver charges \(\$ 2.50\) per mile. (a) Fill in the table with the correct response for the price \(f(x)\) the driver charges for a trip of \(x\) miles. (b) The linear function that gives a rule for the amount charged is \(f(x)=\) (c) Graph this function for the domain \\{0,1,2,3\\} using the set of axes at the right. $$ \begin{array}{c|c} x & f(x) \\ \hline 0 & \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline \end{array} $$
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